Question: $\dfrac{ 6r - 4s }{ 10 } = \dfrac{ -8r - 10t }{ -5 }$ Solve for $r$.
Multiply both sides by the left denominator. $\dfrac{ 6r - 4s }{ {10} } = \dfrac{ -8r - 10t }{ -5 }$ ${10} \cdot \dfrac{ 6r - 4s }{ {10} } = {10} \cdot \dfrac{ -8r - 10t }{ -5 }$ $6r - 4s = {10} \cdot \dfrac { -8r - 10t }{ -5 }$ Reduce the right side. $6r - 4s = {10} \cdot \dfrac{ -8r - 10t }{ -{5} }$ $6r - 4s = -{2} \cdot \left( -8r - 10t \right)$ Distribute the right side $6r - 4s = -{2} \cdot \left( -{8r} - {10t} \right)$ $6r - 4s = {16}r + {20}t$ Combine $r$ terms on the left. ${6r} - 4s = {16r} + 20t$ $-{10r} - 4s = 20t$ Move the $s$ term to the right. $-10r - {4s} = 20t$ $-10r = 20t + {4s}$ Isolate $r$ by dividing both sides by its coefficient. $-{10}r = 20t + 4s$ $r = \dfrac{ 20t + 4s }{ -{10} }$ All of these terms are divisible by $2$ Divide by the common factor and swap signs so the denominator isn't negative. $r = \dfrac{ -{10}t - {2}s }{ {5} }$